Consider x2 - Dy2 = F with all integers. Turns out I found there is a very easy way to solve for x and y modularly.
Given x2 - Dy2 = F where all variables are non-zero integers, with a non-zero integer N for which a residue m exists where m2 = D mod N, and with r, any residue modulo N for which Fr-1 mod N exists then:
2x = r + Fr-1 mod N
2my = r - Fr-1 mod N
It is EASY to derive so you may see if you can figure it out. Or you can see it derived here.
That result gives solutions to x2 - Dy2 = F mod N.
This thing is so simple I find it hard to believe it's a new discovery. So I'm emphasizing that and also still looking for it elsewhere. I think it's simple and cool though, even if it's just another re-discovery.
I do wonder if you could use it with, say, factoring, but haven't noticed anything about which I'm sure. And I think part of me just kind of just wants it to be important, you know?
But then again, I don't know if you could do much with it either. So it's just this thing I have and wonder about once in a while.